What is the next number in my sequence? - Page 2 - Science, Math and Philosophy Forum

The chance a number order n^2 is prime is 1/log[n^2]~1/(2log(n)).

So heuristically one could claim the number of primes of this polynomial above n^2+n+41

is going like Sum[1/(2log(n)) , n=40 to infinity] which diverges. (in fact a bit better by a factor 2 or more even because the above polynomial is delivering only odd numbers)

So a carefully selected polynomial (to avoid stupid factorization ones) should deliver infinite number of primes but with less and less frequency.

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07-12-2017 , 10:24 AM

#29

TomCowley

Pooh-Bah

Join Date: Sep 2004 Posts: 5,649

Quote:

Originally Posted by Zeno

Playing with polynomials was a favorite pastime of mine years ago when taking college math classes. But I noted the bolded - my intuition tells me that the sequence of prime numbers can't be fit to a polynomial expression. But then I think you imply that "any sequence " is non-random repeatability, thus reducible to polynomial expressions. But then is the sequence of prime numbers non-random? I'm probably out of my element here - perhaps lastcard, Masque or this guy Tom can provide some illumination.

lol yeah, sequence in the context of this thread, a finite enumeration. There's not an isomorphism between the set of all possible infinite number sequences and the set of infinite sequences produced by polynomials of degree N (fixed, finite) or less

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07-13-2017 , 09:07 AM

#30

PairTheBoard

Carpal \'Tunnel

Join Date: Dec 2003 Posts: 10,037

Quote:

Originally Posted by TomCowley

If the polynomials must have integer coefficients then I don't think such an isomorphism is possible even with no restriction on the degrees of the polynomials. There are countably many such polynomials of any fixed degree so countably many of unrestricted finite degree; i.e. the countable union of polynomials of fixed degrees. But there are uncountably many sequences, i.e. 2^N where N = set of natural numbers.

PairTheBoard

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07-13-2017 , 04:19 PM

#31

masque de Z

Carpal \'Tunnel

Join Date: Aug 2009 Posts: 9,961

http://mathworld.wolfram.com/Prime-G...olynomial.html

"Legendre showed that there is no rational algebraic function which always gives primes. In 1752, Goldbach showed that no polynomial with integer coefficients can give a prime for all integer values (Nagell 1951, p. 65; Hardy and Wright 1979, pp. 18 and 22). However, there exists a polynomial in 10 variables with integer coefficients such that the set of primes equals the set of positive values of this polynomial obtained as the variables run through all nonnegative integers, although it is really a set of Diophantine equations in disguise (Ribenboim 1991). Jones, Sato, Wada, and Wiens have also found a polynomial of degree 25 in 26 variables whose positive values are exactly the prime numbers (Flannery and Flannery 2000, p. 51). "

That is a good one too.

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